In particular, for the density-dependent Euler equations (REF ), by means of Constantin-E-Titi type
commutators on mollifying kernel, Feireisl-Gwiazda-Gwiazda-Wiedemann [1]} first established the following energy conservation criterion of weak solutions: if the weak solutions satisfy
\(v \in B_{p, \infty }^{\alpha } ((0, T) \times \mathbb {T}^{d} ), \quad \varrho , \varrho v \in B_{p̑{p-2}, \infty }^{\beta } ((0, T) \times \mathbb {T}^{d} ), \quad \pi \in L_{l o c}^{p̑{p-1}} ((0, T) \times \mathbb {T}^{d} ),\) 
